7/30/2023 0 Comments Nicecast retirement![]() ![]() ![]() #Derivation of moment of inertia of a circle full In the sheet metal example above, it is perfectly reasonable to define a mass per unit area. For a long thin rod or chain, it is perfectly reasonable to define a mass per unit length - and is done so often to predict the shape a hanging chain or rope will take. Sure, mathematically, a line is only 1 dimensional, but the math of that 1 dimensional object can be used to predict with incredible accuracy the behavior of a true 3 dimensional object such as a rope or chain. Just like the math for a 2-D object like a circle or square can be used to predict very accurately the behavior of a true 3-D object like a circle or square cut out of sheet metal. Of course in reality there is that 3rd dimension, but it isn't important for the math. It is definitely possible to find the moment of inertia of a circle or square. One more time, while I agree that in physical reality there is no such thing as a solely 2-D object, that in no way whatsoever hinders the development of the mathematics of a 2-D object. The definition of MOI involves a triple integral over 3 space dimensions. Mathematically, if the object is only 2-D, then it acts just like it is a Dirac delta function in that third dimension.Īnd, one more time, treating an object that is very nearly purely 2-D, as in a 1m circle made of sheet metal, that is approximating that very small thickness of the sheet metal as a delta function, can be very accurate. The difference between the two is almost negligible. It is a difference between mathematics and physical reality, but very often the mathematical shortcut is exceptionally good. Other examples is treating the atoms of an ideal gas a point particles yields the perfect gas law, which can be very accurate under specific conditions. When launching a probe from the Earth to Mars, the gravitational influence of Pluto isn't negligible, but treating Pluto as a point mass is accurate enough. ![]() The real art of physics is very often in finding the approximations that make the math significantly simpler, but don't affect the answer a great deal. Physics is replete with examples if you just look for them. Structural Calculators Properties of Areas Centroid.The centroid of a shape represents the point about which the area of the section is evenly distributed. #Derivation of moment of inertia of a circle full. ![]()
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